\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  -25.0 & -1.00 x_{1} & +  8.00 x_{2} & +  3.00 x_{3} & -2.00 x_{4} & -5.00 x_{5} & + 10.00 x_{6}\\
 x_{8}   &  59.0 & +  7.00 x_{1} & -9.00 x_{2} & -7.00 x_{3} & -3.00 x_{4} & -5.00 x_{5} & -8.00 x_{6}\\
 x_{9}   &  20.0 & -6.00 x_{1} &   & -6.00 x_{3} & -6.00 x_{4} & -1.00 x_{5} & +  1.00 x_{6}\\
 x_{10}   &  -8.0 & -1.00 x_{1} &   & -7.00 x_{3} & + 10.00 x_{4} & +  2.00 x_{5} & -3.00 x_{6}\\
 x_{11}   &  21.0 & +  4.00 x_{1} & -2.00 x_{2} & -10.00 x_{3} & +  6.00 x_{4} & +  5.00 x_{5} & -10.00 x_{6}\\
\hline
z    &  0.0 & +  3.00 x_{1} &   & +  5.00 x_{3} & +  2.00 x_{4} & -2.00 x_{5} & +  4.00 x_{6}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{6}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,11$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  1.00 y_{7} & -7.00 y_{8} & +  6.00 y_{9} & +  1.00 y_{10} & -4.00 y_{11}\\
 y_{2}   &  1.0 & -8.00 y_{7} & +  9.00 y_{8} &    &   & +  2.00 y_{11}\\
 y_{3}   &  1.0 & -3.00 y_{7} & +  7.00 y_{8} & +  6.00 y_{9} & +  7.00 y_{10} & + 10.00 y_{11}\\
 y_{4}   &  1.0 & +  2.00 y_{7} & +  3.00 y_{8} & +  6.00 y_{9} & -10.00 y_{10} & -6.00 y_{11}\\
 y_{5}   &  1.0 & +  5.00 y_{7} & +  5.00 y_{8} & +  1.00 y_{9} & -2.00 y_{10} & -5.00 y_{11}\\
 y_{6}   &  1.0 & -10.00 y_{7} & +  8.00 y_{8} & -1.00 y_{9} & +  3.00 y_{10} & + 10.00 y_{11}\\
\hline
z    &  -0 & + 25.00 y_{7} & -59.00 y_{8} & -20.00 y_{9} & +  8.00 y_{10} & -21.00 y_{11}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 4 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.25 & -0.10 y_{4} & -5.35 y_{8} & +  6.60 y_{9} & -0.15 y_{2} & -4.30 y_{11}\\
 y_{10}   &  0.125 & -0.10 y_{4} & +  0.52 y_{8} & +  0.60 y_{9} & -0.03 y_{2} & -0.55 y_{11}\\
 y_{3}   &  1.5 & -0.70 y_{4} & +  7.30 y_{8} & + 10.20 y_{9} & +  0.20 y_{2} & +  5.40 y_{11}\\
 y_{6}   &  0.125 & -0.30 y_{4} & -1.67 y_{8} & +  0.80 y_{9} & +  1.17 y_{2} & +  5.85 y_{11}\\
 y_{5}   &  1.375 & +  0.20 y_{4} & +  9.58 y_{8} & -0.20 y_{9} & -0.57 y_{2} & -2.65 y_{11}\\
 y_{7}   &  0.125  &   & +  1.12 y_{8} &   & -0.12 y_{2} & +  0.25 y_{11}\\
\hline
z    &  4.125 & -0.80 y_{4} & -26.68 y_{8} & -15.20 y_{9} & -3.33 y_{2} & -19.15 y_{11}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.8 & +  0.10 x_{1} & +  0.10 x_{10} & +  0.70 x_{3} & +  0.30 x_{6} & -0.20 x_{5} &   \\
 x_{8}   &  26.675 & +  5.35 x_{1} & -0.52 x_{10} & -7.30 x_{3} & +  1.67 x_{6} & -9.58 x_{5} & -1.12 x_{7}\\
 x_{9}   &  15.2 & -6.60 x_{1} & -0.60 x_{10} & -10.20 x_{3} & -0.80 x_{6} & +  0.20 x_{5} &   \\
 x_{2}   &  3.325 & +  0.15 x_{1} & +  0.03 x_{10} & -0.20 x_{3} & -1.17 x_{6} & +  0.57 x_{5} & +  0.12 x_{7}\\
 x_{11}   &  19.15 & +  4.30 x_{1} & +  0.55 x_{10} & -5.40 x_{3} & -5.85 x_{6} & +  2.65 x_{5} & -0.25 x_{7}\\
\hline
z    &  -4.125 & -1.25 x_{1} & -0.12 x_{10} & -1.50 x_{3} & -0.12 x_{6} & -1.38 x_{5} & -0.12 x_{7}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.8 & +  0.10 x_{1} & +  0.10 x_{10} & +  0.70 x_{3} & +  0.30 x_{6} & -0.20 x_{5} &   \\
 x_{8}   &  26.675 & +  5.35 x_{1} & -0.52 x_{10} & -7.30 x_{3} & +  1.67 x_{6} & -9.58 x_{5} & -1.12 x_{7}\\
 x_{9}   &  15.2 & -6.60 x_{1} & -0.60 x_{10} & -10.20 x_{3} & -0.80 x_{6} & +  0.20 x_{5} &   \\
 x_{2}   &  3.325 & +  0.15 x_{1} & +  0.03 x_{10} & -0.20 x_{3} & -1.17 x_{6} & +  0.57 x_{5} & +  0.12 x_{7}\\
 x_{11}   &  19.15 & +  4.30 x_{1} & +  0.55 x_{10} & -5.40 x_{3} & -5.85 x_{6} & +  2.65 x_{5} & -0.25 x_{7}\\
\hline
z    &  1.6 & +  3.20 x_{1} & +  0.20 x_{10} & +  6.40 x_{3} & +  4.60 x_{6} & -2.40 x_{5} &   \\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.8 & +  0.10 x_{1} & +  0.10 x_{10} & +  0.70 x_{3} & +  0.30 x_{6} & -0.20 x_{5} &   \\
 x_{8}   &  26.675 & +  5.35 x_{1} & -0.52 x_{10} & -7.30 x_{3} & +  1.67 x_{6} & -9.58 x_{5} & -1.12 x_{7}\\
 x_{9}   &  15.2 & -6.60 x_{1} & -0.60 x_{10} & -10.20 x_{3} & -0.80 x_{6} & +  0.20 x_{5} &   \\
 x_{2}   &  3.325 & +  0.15 x_{1} & +  0.03 x_{10} & -0.20 x_{3} & -1.17 x_{6} & +  0.57 x_{5} & +  0.12 x_{7}\\
 x_{11}   &  19.15 & +  4.30 x_{1} & +  0.55 x_{10} & -5.40 x_{3} & -5.85 x_{6} & +  2.65 x_{5} & -0.25 x_{7}\\
\hline
z    &  1.6 & +  3.20 x_{1} & +  0.20 x_{10} & +  6.40 x_{3} & +  4.60 x_{6} & -2.40 x_{5} &   \\
\end{array}\]


 $ x_{1} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  1.0303030303 & -0.02 x_{9} & +  0.09 x_{10} & +  0.55 x_{3} & +  0.29 x_{6} & -0.20 x_{5} &   \\
 x_{8}   &  38.9962121212 & -0.81 x_{9} & -1.01 x_{10} & -15.57 x_{3} & +  1.03 x_{6} & -9.41 x_{5} & -1.12 x_{7}\\
 x_{1}   &  2.30303030303 & -0.15 x_{9} & -0.09 x_{10} & -1.55 x_{3} & -0.12 x_{6} & +  0.03 x_{5} &   \\
 x_{2}   &  3.67045454545 & -0.02 x_{9} & +  0.01 x_{10} & -0.43 x_{3} & -1.19 x_{6} & +  0.58 x_{5} & +  0.12 x_{7}\\
 x_{11}   &  29.053030303 & -0.65 x_{9} & +  0.16 x_{10} & -12.05 x_{3} & -6.37 x_{6} & +  2.78 x_{5} & -0.25 x_{7}\\
\hline
z    &  8.9696969697 & -0.48 x_{9} & -0.09 x_{10} & +  1.45 x_{3} & +  4.21 x_{6} & -2.30 x_{5} &   \\
\end{array}\]


 $ x_{3} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  1.8431372549 & -0.07 x_{9} & +  0.06 x_{10} & -0.35 x_{1} & +  0.25 x_{6} & -0.19 x_{5} &   \\
 x_{8}   &  15.7965686275 & +  0.72 x_{9} & -0.10 x_{10} & + 10.07 x_{1} & +  2.25 x_{6} & -9.72 x_{5} & -1.12 x_{7}\\
 x_{3}   &  1.49019607843 & -0.10 x_{9} & -0.06 x_{10} & -0.65 x_{1} & -0.08 x_{6} & +  0.02 x_{5} &   \\
 x_{2}   &  3.02696078431 & +  0.02 x_{9} & +  0.04 x_{10} & +  0.28 x_{1} & -1.16 x_{6} & +  0.57 x_{5} & +  0.12 x_{7}\\
 x_{11}   &  11.1029411765 & +  0.53 x_{9} & +  0.87 x_{10} & +  7.79 x_{1} & -5.43 x_{6} & +  2.54 x_{5} & -0.25 x_{7}\\
\hline
z    &  11.137254902 & -0.63 x_{9} & -0.18 x_{10} & -0.94 x_{1} & +  4.10 x_{6} & -2.27 x_{5} &   \\
\end{array}\]


 $ x_{6} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.34462511292 & -0.04 x_{9} & +  0.10 x_{10} & -0.00 x_{1} & -0.05 x_{11} & -0.07 x_{5} & -0.01 x_{7}\\
 x_{8}   &  20.3952122855 & +  0.93 x_{9} & +  0.26 x_{10} & + 13.30 x_{1} & -0.41 x_{11} & -8.66 x_{5} & -1.23 x_{7}\\
 x_{3}   &  1.32971996387 & -0.11 x_{9} & -0.07 x_{10} & -0.76 x_{1} & +  0.01 x_{11} & -0.02 x_{5} & +  0.00 x_{7}\\
 x_{2}   &  0.654923215899 & -0.09 x_{9} & -0.15 x_{10} & -1.39 x_{1} & +  0.21 x_{11} & +  0.03 x_{5} & +  0.18 x_{7}\\
 x_{6}   &  2.0460704607 & +  0.10 x_{9} & +  0.16 x_{10} & +  1.44 x_{1} & -0.18 x_{11} & +  0.47 x_{5} & -0.05 x_{7}\\
\hline
z    &  19.522131888 & -0.23 x_{9} & +  0.48 x_{10} & +  4.94 x_{1} & -0.76 x_{11} & -0.35 x_{5} & -0.19 x_{7}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.34419817471 & -0.04 x_{9} & +  0.10 x_{10} & +  0.00 x_{2} & -0.05 x_{11} & -0.07 x_{5} & -0.01 x_{7}\\
 x_{8}   &  26.6818774446 & +  0.04 x_{9} & -1.16 x_{10} & -9.60 x_{2} & +  1.64 x_{11} & -8.40 x_{5} & +  0.48 x_{7}\\
 x_{3}   &  0.970664928292 & -0.05 x_{9} & +  0.01 x_{10} & +  0.55 x_{2} & -0.10 x_{11} & -0.03 x_{5} & -0.09 x_{7}\\
 x_{1}   &  0.472620599739 & -0.07 x_{9} & -0.11 x_{10} & -0.72 x_{2} & +  0.15 x_{11} & +  0.02 x_{5} & +  0.13 x_{7}\\
 x_{6}   &  2.72490221643 & +  0.00 x_{9} & +  0.01 x_{10} & -1.04 x_{2} & +  0.04 x_{11} & +  0.50 x_{5} & +  0.14 x_{7}\\
\hline
z    &  21.8591916558 & -0.56 x_{9} & -0.05 x_{10} & -3.57 x_{2} & +  0.01 x_{11} & -0.25 x_{5} & +  0.45 x_{7}\\
\end{array}\]


 $ x_{7} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  2.22664359862 & -0.04 x_{9} & +  0.10 x_{10} & -0.07 x_{2} & -0.03 x_{11} & -0.07 x_{5} & +  0.12 x_{3}\\
 x_{8}   &  31.669550173 & -0.24 x_{9} & -1.11 x_{10} & -6.78 x_{2} & +  1.11 x_{11} & -8.57 x_{5} & -5.14 x_{3}\\
 x_{7}   &  10.3044982699 & -0.58 x_{9} & +  0.11 x_{10} & +  5.82 x_{2} & -1.09 x_{11} & -0.34 x_{5} & -10.62 x_{3}\\
 x_{1}   &  1.79930795848 & -0.14 x_{9} & -0.09 x_{10} & +  0.03 x_{2} & +  0.01 x_{11} & -0.02 x_{5} & -1.37 x_{3}\\
 x_{6}   &  4.15570934256 & -0.08 x_{9} & +  0.02 x_{10} & -0.23 x_{2} & -0.11 x_{11} & +  0.45 x_{5} & -1.47 x_{3}\\
\hline
z    &  26.4740484429 & -0.82 x_{9} & -0.00 x_{10} & -0.96 x_{2} & -0.48 x_{11} & -0.41 x_{5} & -4.75 x_{3}\\
\end{array}\]
Final Dictionary
Solution:  26.4740484429
Num Pivots:  6
\end{document}
